Measures from Dixmier Traces and Zeta Functions

Details Measures from Dixmier Traces and Zeta Functions Measures from Dixmier Traces and Zeta Functions

2009-05-08

arxiv.org/abs/0905.1172v2

J. Funct. Anal. (2010)

Mathematics

Functional Analysis

28 pages, submitted to J. Funct. Anal.

Scientific paper

10.1016/j.jfa.2010.06.012

For essentially bounded functions on a (closed) compact Riemannian manifold, the noncommutative residue and the Dixmier trace formulation of the noncommutative integral are shown to equate to a multiple of the Lebesgue integral. The identifications are shown to continue to, and be sharp at, square integrable functions. To do better than square integrable, symmetrised noncommutative residue and Dixmier trace formulas are introduced, for which the identifications are shown to continue to $L^{1+\epsilon}$-spaces, $\epsilon > 0$. However, a failure is shown for the Dixmier trace formulation at integrable functions. The (symmetrised) noncommutative residue and Dixmier trace formulas diverge at this point. It is shown the noncommutative residue remains finite and recovers the Lebesgue integral for any integrable function while the Dixmier trace expression can diverge. The results show the claim (in the monograph "Elements of Noncommutative Geometry", Birkhauser, 2001), that the identification on smooth functions obtained using Connes' Trace Theorem can be extended to any integrable function, is false.

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